Measure-preserving rank one transformations

Ryzhikov, V.

doi:10.1090/mosc/310

Measure-preserving rank one transformations
HTML articles powered by AMS MathViewer

by V. V. Ryzhikov
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2020, 229-259

DOI: https://doi.org/10.1090/mosc/310

Published electronically: March 15, 2021

PDF | Request permission

Abstract:

Rank 1 transformations serve as a source of examples in ergodic theory showing a variety of algebraic, asymptotic, and spectral properties of dynamical systems. The properties of a rank one transformation are closely related to the structure of the semigroup of weak limits of its powers. In this vein, known and new constructions of transformations are studied.

References

O. N. Ageev, The generic automorphism of a Lebesgue space conjugate to a $G$-extension for any finite abelian group $G$, Dokl. Akad. Nauk 374 (2000), no. 4, 439–442 (Russian). MR 1798480
A. I. Bashtanov, Generic mixing transformations are rank 1, Math. Notes 93 (2013), no. 1-2, 209–216. Translation of Mat. Zametki 93 (2013), no. 2, 163–171. MR 3205962, DOI 10.1134/S0001434613010227
A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Dokl. Akad. Nauk SSSR 259 (1981), no. 3, 526–529 (Russian). MR 625756
A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 81–106 (Russian). MR 0219697
A. Yu. Kushnir and V. V. Ryzhikov, Weak closures of ergodic actions, Mat. Zametki 100 (2016), no. 6, 847–854 (Russian, with Russian summary); English transl., Math. Notes 101 (2017), no. 1-2, 277–283. MR 3588909, DOI 10.4213/mzm10965
M. S. Lobanov and V. V. Ryzhikov, Special weak limits and the simple spectrum of the tensor products of flows, Mat. Sb. 209 (2018), no. 5, 62–73 (Russian, with Russian summary); English transl., Sb. Math. 209 (2018), no. 5, 660–671. MR 3795151, DOI 10.4213/sm8932
Yu. A. Neretin, Symmetry categories and infinite groups, URSS, Moscow, 1998. (Russian)
V. I. Oseledec, An automorphism with simple and continuous spectrum and without the group property, Mat. Zametki 5 (1969), 323–326 (Russian). MR 257323
V. I. Oseledec, Two nonisomorphic dynamic systems with identical simple continuous spectrum, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 75–79 (Russian). MR 0283173
V. V. Ryzhikov, Mixing, rank and minimal self-joining of actions with invariant measure, Mat. Sb. 183 (1992), no. 3, 133–160 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 75 (1993), no. 2, 405–427. MR 1180921, DOI 10.1070/SM1993v075n02ABEH003391
V. V. Ryzhikov, Wreath products of tensor products, and a stochastic centralizer of dynamical systems, Mat. Sb. 188 (1997), no. 2, 67–94 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 2, 237–263. MR 1453260, DOI 10.1070/SM1997v188n02ABEH000202
V. V. Ryzhikov, The Rokhlin problem on multiple mixing in the class of actions of positive local rank, Funktsional. Anal. i Prilozhen. 34 (2000), no. 1, 90–93 (Russian); English transl., Funct. Anal. Appl. 34 (2000), no. 1, 73–75. MR 1756740, DOI 10.1007/BF02467072
V. V. Ryzhikov, On the spectral and mixing properties of rank-1 constructions in ergodic theory, Dokl. Akad. Nauk 409 (2006), no. 4, 448–450 (Russian). MR 2350974
V. V. Ryzhikov and Zh.-P. Tuveno, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions, Funktsional. Anal. i Prilozhen. 40 (2006), no. 3, 85–89 (Russian); English transl., Funct. Anal. Appl. 40 (2006), no. 3, 237–240. MR 2265691, DOI 10.1007/s10688-006-0038-8
V. V. Ryzhikov, Factors, rank, and embedding of a generic $\Bbb Z^n$-action in an $\Bbb R^n$-flow, Uspekhi Mat. Nauk 61 (2006), no. 4(370), 197–198 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 4, 786–787. MR 2278846, DOI 10.1070/RM2006v061n04ABEH004351
V. V. Ryzhikov, Weak limits of powers, the simple spectrum of symmetric products, and mixing constructions of rank 1, Mat. Sb. 198 (2007), no. 5, 137–159 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 5-6, 733–754. MR 2354530, DOI 10.1070/SM2007v198n05ABEH003857
V. V. Ryzhikov, Bounded ergodic constructions, disjointness, and weak limits of powers, Trans. Moscow Math. Soc. , posted on (2013), 165–171. MR 3235793, DOI 10.1090/s0077-1554-2014-00214-4
V. V. Ryzhikov, Ergodic homoclinic groups, Sidon constructions and Poisson suspensions, Trans. Moscow Math. Soc. , posted on (2014), 77–85. MR 3308601, DOI 10.1090/s0077-1554-2014-00227-2
V. V. Ryzhikov and J.-P. Thouvenot, On the centralizer of an infinite mixing rank-one transformation, Funct. Anal. Appl. 49 (2015), no. 3, 230–233. Translation of Funktsional. Anal. i Prilozhen. 49 (2015), no. 3, 88–91. MR 3402415, DOI 10.1007/s10688-015-0111-2
V. V. Ryzhikov, Thouvenot’s isomorphism problem for tensor powers of ergodic flows, Mat. Zametki 104 (2018), no. 6, 912–917 (Russian, with Russian summary); English transl., Math. Notes 104 (2018), no. 5-6, 900–904. MR 3881780, DOI 10.4213/mzm11987
V. V. Ryzhikov, Weakly homoclinic groups of ergodic actions, Trans. Moscow Math. Soc. 80 (2019), 83–94. MR 4082861, DOI 10.1090/mosc/289
V. V. Ryzhikov, Weak closure of infinite actions of rank 1, joinings and spectrum, Mat. Zametki 106 (2019), no. 6, 894–903 (Russian, with Russian summary); English transl., Math. Notes 106 (2019), no. 5-6, 957–965. MR 4045675, DOI 10.4213/mzm12303
A. M. Stëpin, Spectral properties of generic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 801–834, 879 (Russian). MR 864178
A. M. Stepin and A. M. Eremenko, Nonuniqueness of an inclusion in a flow and the vastness of a centralizer for a generic measure-preserving transformation, Mat. Sb. 195 (2004), no. 12, 95–108 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 11-12, 1795–1808. MR 2138483, DOI 10.1070/SM2004v195n12ABEH000867
S. V. Tikhonov, Complete metric on the set of mixing transformations, Uspekhi Mat. Nauk 62 (2007), no. 1(373), 209–210 (Russian); English transl., Russian Math. Surveys 62 (2007), no. 1, 193–195. MR 2354213, DOI 10.1070/RM2007v062n01ABEH004392
Terrence M. Adams, Smorodinsky’s conjecture on rank-one mixing, Proc. Amer. Math. Soc. 126 (1998), no. 3, 739–744. MR 1443143, DOI 10.1090/S0002-9939-98-04082-9
Oleg Ageev, The homogeneous spectrum problem in ergodic theory, Invent. Math. 160 (2005), no. 2, 417–446. MR 2138072, DOI 10.1007/s00222-004-0422-z
Oleg N. Ageev, Spectral rigidity of group actions: applications to the case $\textrm {gr}\langle t,s;\ ts=st^2\rangle$, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1331–1338. MR 2199176, DOI 10.1090/S0002-9939-05-08380-2
Steve Alpern, Conjecture: in general a mixing transformation is not two-fold mixing, Ann. Probab. 13 (1985), no. 1, 310–313. MR 770646
J. R. Baxter, A class of ergodic transformations having simple spectrum, Proc. Amer. Math. Soc. 27 (1971), 275–279. MR 276440, DOI 10.1090/S0002-9939-1971-0276440-2
J. Bourgain, On the spectral type of Ornstein’s class one transformations, Israel J. Math. 84 (1993), no. 1-2, 53–63. MR 1244658, DOI 10.1007/BF02761690
J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math. 120 (2013), 105–130. MR 3095150, DOI 10.1007/s11854-013-0016-z
R. V. Chacon, A geometric construction of measure preserving transformations, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 335–360. MR 0212158
R. V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc. 22 (1969), 559–562. MR 247028, DOI 10.1090/S0002-9939-1969-0247028-5
J. Chaika and B. Kra, A prime system with many self-joinings, arXiv:1902.02421.
J. Chaika and D. Davis, The typical measure preserving transformation is not an interval exchange transformation, arXiv:1812.10425.
Darren Creutz and Cesar E. Silva, Mixing on rank-one transformations, Studia Math. 199 (2010), no. 1, 43–72. MR 2652597, DOI 10.4064/sm199-1-4
Alexandre I. Danilenko, $(C,F)$-actions in ergodic theory, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, Basel, 2008, pp. 325–351. MR 2402408, DOI 10.1007/978-3-7643-8608-5_{6}
Alexandre I. Danilenko, A survey on spectral multiplicities of ergodic actions, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 81–117. MR 3009104, DOI 10.1017/S0143385711000800
El Houcein El Abdalaoui, Mariusz Lemańczyk, and Thierry de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal. 266 (2014), no. 1, 284–317. MR 3121731, DOI 10.1016/j.jfa.2013.09.005
Matthew Foreman, Daniel J. Rudolph, and Benjamin Weiss, The conjugacy problem in ergodic theory, Ann. of Math. (2) 173 (2011), no. 3, 1529–1586. MR 2800720, DOI 10.4007/annals.2011.173.3.7
E. Glasner, B. Host, and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math. 78 (1992), no. 1, 131–142. MR 1194963, DOI 10.1007/BF02801575
Bernard Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math. 76 (1991), no. 3, 289–298. MR 1177346, DOI 10.1007/BF02773866
Élise Janvresse, Thierry de la Rue, and Valery Ryzhikov, Around King’s rank-one theorems: flows and $\Bbb Z^n$-actions, Dynamical systems and group actions, Contemp. Math., vol. 567, Amer. Math. Soc., Providence, RI, 2012, pp. 143–161. MR 2931916, DOI 10.1090/conm/567/11233
É. Janvresse, A. A. Prikhod’ko, T. de la Rue, and V. V. Ryzhikov, Weak limits of powers of Chacon’s automorphism, Ergodic Theory Dynam. Systems 35 (2015), no. 1, 128–141. MR 3294294, DOI 10.1017/etds.2013.49
Élise Janvresse, Emmanuel Roy, and Thierry de la Rue, Poisson suspensions and Sushis, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 6, 1301–1334 (English, with English and French summaries). MR 3742194, DOI 10.24033/asens.2346
Andrés del Junco, Transformations with discrete spectrum are stacking transformations, Canadian J. Math. 28 (1976), no. 4, 836–839. MR 414822, DOI 10.4153/CJM-1976-080-3
A. del Junco, M. Rahe, and L. Swanson, Chacon’s automorphism has minimal self-joinings, J. Analyse Math. 37 (1980), 276–284. MR 583640, DOI 10.1007/BF02797688
A. del Junco and D. J. Rudolph, A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 229–247. MR 896795, DOI 10.1017/S0143385700003977
Andrés del Junco, A simple map with no prime factors, Israel J. Math. 104 (1998), 301–320. MR 1622315, DOI 10.1007/BF02897068
Anatole Katok, Combinatorial constructions in ergodic theory and dynamics, University Lecture Series, vol. 30, American Mathematical Society, Providence, RI, 2003. MR 2008435, DOI 10.1090/ulect/030
Steven Arthur Kalikow, Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 237–259. MR 766104, DOI 10.1017/S014338570000242X
Jonathan King, The commutant is the weak closure of the powers, for rank-$1$ transformations, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 363–384. MR 863200, DOI 10.1017/S0143385700003552
Jonathan L. King, Joining-rank and the structure of finite rank mixing transformations, J. Analyse Math. 51 (1988), 182–227. MR 963154, DOI 10.1007/BF02791123
Jonathan L. King, Ergodic properties where order $4$ implies infinite order, Israel J. Math. 80 (1992), no. 1-2, 65–86. MR 1248927, DOI 10.1007/BF02808154
Jonathan L. F. King, The generic transformation has roots of all orders. part 2, Colloq. Math. 84/85 (2000), no. part 2, 521–547. Dedicated to the memory of Anzelm Iwanik. MR 1784212, DOI 10.4064/cm-84/85-2-521-547
Ivo Klemes and Karin Reinhold, Rank one transformations with singular spectral type, Israel J. Math. 98 (1997), 1–14. MR 1459845, DOI 10.1007/BF02937326
Donald S. Ornstein, On invariant measures, Bull. Amer. Math. Soc. 66 (1960), 297–300. MR 146350, DOI 10.1090/S0002-9904-1960-10478-8
Donald S. Ornstein, On the root problem in ergodic theory, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 347–356. MR 0399415
A. A. Prikhod′ko and V. V. Ryzhikov, Disjointness of the convolutions for Chacon’s automorphism. part 1, Colloq. Math. 84/85 (2000), no. part 1, 67–74. Dedicated to the memory of Anzelm Iwanik. MR 1778840, DOI 10.4064/cm-84/85-1-67-74
Daniel J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97–122. MR 555301, DOI 10.1007/BF02791063
Thierry de la Rue and José de Sam Lazaro, Une transformation générique peut être insérée dans un flot, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 1, 121–134 (French, with English and French summaries). MR 1959844, DOI 10.1016/S0246-0203(02)00012-2
V. V. Ryzhikov, Stochastic intertwinings and multiple mixing of dynamical systems, J. Dynam. Control Systems 2 (1996), no. 1, 1–19. MR 1377426, DOI 10.1007/BF02259620
V. V. Ryzhikov, Homogeneous spectrum, disjointness of convolutions, and mixing properties of dynamical systems, Selected Russ. Math. 1 (1999), 13–24; arXiv:1206.6093.
V. V. Ryzhikov, Weak closure theorem for double staircase actions, arXiv:1108.0568.
V. V. Ryzhikov, On mixing constructions with algebraic spacers, arXiv:1108.1508.
V. V. Ryzhikov, On mixing of staircase transformations, arXiv:1108.3522.
V. V. Ryzhikov, On disjointness of mixing rank one actions, arXiv:1109.0671.
V. V. Ryzhikov, Bounded ergodic constructions, disjointness, and weak limits of powers, Trans. Moscow Math. Soc. , posted on (2013), 165–171. MR 3235793, DOI 10.1090/s0077-1554-2014-00214-4
V. V. Ryzhikov, Chacon’s type ergodic transformations with unbounded arithmetic spacers, arXiv:1311.4524.
S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. MR 0120127
Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121–1174. MR 3171761, DOI 10.4007/annals.2014.179.3.7